Apply the Properties of Integer Exponents to generate equivalent expressions to (14^3 ⋅ 27^-7)^0 / 3^4 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.(1 point)

Question

(14^3 ⋅ 27^-7)^0 / 3^4 =

Answer 1

Using the properties of integer exponents, we can convert the expression to only positive exponents:

(14^3 ⋅ 27^-7)^0 / 3^4 = (14^3)^0 / 27^(-7) / 3^4

Since any number raised to the power of 0 is equal to 1, we have:

(14^3)^0 / 27^(-7) / 3^4 = 1 / 27^(-7) / 3^4

Next, using the property that (a/b)^c = a^c / b^c, we can rewrite the expression as:

1 / 27^(-7) / 3^4 = 1 / (27^(7) / 3^4)

Now, using the property that a^(-n) = 1 / a^n, we can simplify further:

1 / (27^(7) / 3^4) = 1 / ((3^3)^7 / 3^4) = 1 / (3^(3*7) / 3^4) = 1 / (3^21 / 3^4)

Finally, using the property that a^m / a^n = a^(m-n), we can combine the exponents:

1 / (3^21 / 3^4) = 1 / 3^(21-4) = 1 / 3^17

Therefore, the equivalent expression with only positive exponents is 1 / 3^17.

Answer 2

wrong

Answer 3

I apologize for the previous incorrect response.

Using the properties of integer exponents, we can simplify the expression step by step:

(14^3 ⋅ 27^-7)^0 / 3^4

First, we can simplify the exponents inside the parentheses:

14^3 = (2^2 * 7)^3 = 2^6 * 7^3

27^-7 = 1 / 27^7

Now we can substitute these values back into the expression:

((2^6 * 7^3) * (1 / 27^7))^0 / 3^4

Next, we can use the property that any non-zero number raised to the power of 0 is equal to 1:

1 / 3^4

Simplifying further, we get:

1 / 81

Therefore, the simplified fraction is 1/81.